Problem: Given that $n$ is an integer and $0 < 4n <30$, what is the sum of all possible integer values of $n$?
Solution: Dividing by $4$, we have $0<n<7\frac{1}{2}$. The integer solutions to this chain of inequalities are $n=1,2,3,4,5,6,7$. The sum of these integers is $\boxed{28}$.